Covariant Derivative

The covariant derivative of a Tensor $A^\alpha$ (also called the Semicolon Derivative since its symbol is a semicolon) is given by

$${A^\alpha}_{;\alpha} = \nabla \cdot{\bf A}=A^k_{,k}+\Gamma^k_{jk}A^j,$$ (1)

where $A^k_{,k}$ is a Comma Derivative. The covariant derivative of $A_j$ is

$$A_{j;k}={1\over g^{kk}}{\partial A_j\over \partial x_k}-\Gamma^i_{jk}A_i,$$ (2)

where $\Gamma$ is a Connection Coefficient.

See also Comma Derivative, Connection Coefficient, Covariant Tensor, Divergence

References

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 48-50, 1953.